## Transcribed Text

la) Prove that the infinite product
converges normally in C. thus f(z) is an entire function.
b) Relate f(z) to the function
g(z)=sinz
Hint: Use that
2) Consider
for Res>0
a) Prove that, as n 100 the partial sums
converge uniformly on every compact subset K of the half-plane
H={s : Res>0}
This holds although the series defining h(s) does not converge absolutely unless
Res > 1.
Thus, h(s) is holomorphic in the half-plane H.
b) Prove the
h(s)=(1-21-s)s(s)forRes>1
Hint: Consider (s) h(s).
c) Prove that
h(s)21-1>0 for
s>0.
d) Prove that
(s) <0 for 0<8<1
Here (s) denotes the analytic continuation of the function
1
Res>1.
3) Let aj denote a sequence of complex numbers. Then
(0.1)
is called a Dirichlet series.
Assume that Res>1. Prove: If the sequence aj is bounded then the series
(0.1) converges absolutely and defines a function f(s) which is holomorphic in
the half-plane
H={s: Res>1}.
4) Let aj and bk denote two bounded sequences of complex numbers and let
for
Prove that
for Res>1
(0.2)
where
jk=n
The sum defining Cn is the sum of all products ajbk where j and k are positive
integers with jk = n.
Prove that the series
for
that
holds
1) Let Pj denote the j-th prime, i.e., p1=2,p2= 3,p3 = 5. etc. Determine the
following limits:
and
lim
Hint: Recall what we learned about
2) Let denote the straight line with parametrization
s(t)=c+it,-00<t<00
Prove the formula
if O<a<1 a="">1
for c>0.
Hint: This is a result in Chapter 7 of [Stein, Shakarchi]. Carry out the
details
3) Recall the reflection formula for the T-function
for 8EC\Z.
Use the formula to prove
for ter.
4a) Use the Prime Number Theorem to prove that
x-00
In r
b) Let Pj denote the j-th prime number. Prove that
Hint: Use the PNT for x = p, and use a).
</a<1></t<00

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